Re: -- Kuratowski Ordered Pair ( is the Kuratowski set, an asymmetrical set - here a simple result based on Hamiltons ordered couples or temporal quantities or directed quantities)

An after-math
The euro-sign EURO did not turn out.
Written in the window, where You insert Your message with Google-
groups,
it is displayed, but not, when google-groups puts the mesage up into
th group.
(Here a try with notepad: EURO )

I wrote about zero as an involution.
0 times 0 = 0,
so it is an involution in a space with just on element, where 0 = 1.
Otherwize it's never an involution, as 0 * 0 =|= 1.

Black magician is refering to cheating magic, an expression from
people,
who were made frightened with darkness at nights.

Wessel's introduction of direction into Euclid's geometry is
introducing
a temporal concept into a static geometry.
There was dynamic geometry already before the Elements:
Hippias of Elis' Qudratrix, Archytas 'fingertip'.

Sticking to a static geometry we can advance it with the concept
of fixed alignment in space.
An invariant of rotations in spatial space is an axis, like in
Foucault's
pendulum or in a gyro. Just the axis, without the orientation of the
rotation,
gives us a bi-direction, an alignment, fix in space, pointing to and
from
a fix-star ( distance / speed = very long duration).
For every diameter of a sphere in Euclid's geometry there is one
corresponding
diameter in any other sphere, a parallel one, so all straight lines
fall into
equivalence classes, in a modern expression. Each equivalence class
can be
thought of as one alignment.

Two straight lines through a common point are sharing two pairs of
opposite angles.
In this way an angle can be thought of as a difference of alignment.
Two halflines, sharing a point as their common ends, display two
angles. Without
further information, the smaller one is taken as the angle. In a
concav quadrangle,
with the form of an arrowhead, one takes the bigger angle when talking
about the
inner angle of the concave corner, as here it is referenced.


Three rotational axes, three points can reference any aligment in
space, with an
origin at Your position. Four points, four axes gives redundant
information for
error correction, best arranged as a tetrahedron. Every pair of
opposite edges,
taken together with their inner line of shortest distance gives
perpendicularity.
Four laser gyros, four equal angled triangles can be arranged as a
tetra as well,
giving the optimal platform of reference.

Now back to Kuratowski.

With friendly greetings
Hero
.

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